2. 21
Case Two risky assets
Following the same procedure as in case 1, we allocate
fraction of fund to risky asset 1 and
fraction
of fund to risky asset 2. The rate of return on portfolio is
and its mean and variance are
Both the mean and variance are affected by allocation weights
and
. The relation between
and
that describes the PPC can be derived as follows.
(i) solve the mean equation for
to derive
(ii) Substitute these expressions into the variance equation
Though this expression is complicated and is not very useful for our purpose, it shows that the PPC is a
hyperbola in
and
.
Two cases need a special attention. The equation for the PPC in the case of a perfectly correlated returns
(D=1) becomes a simple equation.
Perfect Positive Correlation (D=+1)
Perfect Negative Correlation (D=-1).
(i)
22
(ii)
If
, the portfolio has a zero risk (
).
Proof.
(A) Perfect Positive Correlation (D=+1).
The portfolio possibility curve is a straight line. In this case, the variance equation becomes
which gives
Solving this for
, we can write
Substituting this into the equation for the mean return we have
And this is a linear equation in the
and
space, and the PPC is a straight line connecting point A and
point B.
(B) Perfect Negative Correlation (D=-1).
In this case, the variance of the portfolio becomes
23
We need to consider two cases because
value
can be positive or negative depending on the allocation
:
(i) positive case:
(ii) negative case:
Note first that, if
, then the portfolio has a zero risk (
. Substituting the solution for
from these
) and the mean return is
equations into the mean return
equations, we derive two segments of the portfolio possibility curves:
(i)
(ii)
QED
Numerical method of finding PPC in a general case
When the returns are not perfectly correlated, one may find the PPC numerically.
Method 1.
Recall that, when short sales are not allowed,
as one wishes. For each value of
, compute
must be in a unit interval. Divide the unit interval as finely
and
by
Remark: For the PPC in terms of the mean and variance of final wealth, simply compute
Method 2.
For a given
in the interval from low mean and high mean of the two stocks, compute
hyperbola equation presented above:
from the
24
Method 3. Excel Solver
For a given mean
, minimize
with respect to
subject to the constraint of mean equation. This is not
useful at all as we have the simple formula in method 2.
Example
The figure below shows the portfolio possibility curves of two assets when short sales are not allowed. The
mean and variance of the two assets and each curve represents different values of correlation coefficient
ranging from -1 to +1. These two assets are taken from one of the examples the in textbook.
:
F
S&P
0.140
0.185
Small Stock
0.17
0.24
Remarks
(i) As shown above, the PPC is a straight line when D=+1. The risk of the portfolio cannot be smaller than
the risk of either asset.
(ii) As D becomes smaller, the PPC bends leftward, and some portfolios can have a smaller risk than the risk
of risky assets. How is this possible?
(ii) If D=-1, there is a portfolio that has a zero risk (
a proper mix makes it flat.
). As the two assets move in the opposite direction,
25
Minimum Risk Portfolio
An extremely risk averse agent may care only about the risk and may choose a portfolio that has the smallest
risk. The portfolio of minimum risk is defined by the allocation weight
Proof. To find the portfolio of minimum risk, take a derivative of
with respect to
, set it to zero and
solve for
The solution for
is the fraction given above.
Efficient Frontier
The portfolio possibility curve above the minimum risk point is called the efficient frontier. The optimum
portfolio will be selected only on the efficient frontier. As long as agents prefer a higher expected return and
a lower risk, then the portion of the possibility curve below the minimum risk will never be chosen.
The efficient frontier is always concave. The efficient frontier cannot have a convex portion because a
portfolio consisting of portfolios on the convex portion can dominate. See Chapter 5 for various examples.
26
Optimum Portfolio - Maximum Expected Utility
If the objective of the portfolio selection is to minimize the risk (F), then the minimum risk portfolio shown
above is the answer.
If the utility function is a CARA or a quadratic function, then we need to find the portfolio that maximizes
the expected utility.
and
in the figures below are the expected utility maximizing mean and risk. This
is the tangent point of the PPC and one of the indifference curves. Given these optimum values, we need to
find the allocation weight
that produce the optimum mean and risk.
We will consider the expected utility of CARA utility function under the normality. The expected utility is
where
,
We have shown that the maximization of this expected utility is equivalent to the maximization of
. The first order condition is
From the expressions of mean and variance, we have
27
Substituting these into the first order condition, we can derive
There is no guarantee that this solution is in the unit interval. If short sales are not allowed, the weights have
to be in the unit interval. Therefore, if this solution is greater than 1, we take 1 as the solution. If this solution
is negative, we take zero as the solution. If short sales are allowed,
1) or greater than 1 (short sales of asset 2).
can be negative (short sales of asset